# Properties

 Label 60.3.f.b Level $60$ Weight $3$ Character orbit 60.f Analytic conductor $1.635$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 60.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.63488158616$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.389136420864.4 Defining polynomial: $$x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( -1 + \beta_{4} ) q^{4} + ( 1 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{5} + ( -1 + \beta_{3} ) q^{6} + ( 2 \beta_{1} + 4 \beta_{2} - \beta_{6} + \beta_{7} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( -1 + \beta_{4} ) q^{4} + ( 1 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{5} + ( -1 + \beta_{3} ) q^{6} + ( 2 \beta_{1} + 4 \beta_{2} - \beta_{6} + \beta_{7} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{8} + 3 q^{9} + ( -5 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{10} + ( -2 \beta_{3} - 4 \beta_{4} - \beta_{6} - \beta_{7} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{12} + ( -2 \beta_{1} - \beta_{6} + \beta_{7} ) q^{13} + ( 2 + 4 \beta_{3} + 2 \beta_{4} ) q^{14} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{15} + ( -7 - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{16} + ( -2 \beta_{1} - 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + 3 \beta_{1} q^{18} + ( -4 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{19} + ( 6 - 6 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{20} + ( 8 + 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{21} + ( 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{22} + ( 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{23} + ( -1 - 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{24} + ( 5 - 2 \beta_{3} + 4 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{25} + ( 10 - 2 \beta_{4} ) q^{26} + 3 \beta_{2} q^{27} + ( -4 \beta_{1} - 12 \beta_{2} - 2 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{28} + ( -26 + 6 \beta_{3} + 3 \beta_{6} + 3 \beta_{7} ) q^{29} + ( -8 - 2 \beta_{1} - 6 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{30} + ( 8 \beta_{3} + 8 \beta_{4} ) q^{31} + ( -2 \beta_{1} + 18 \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{32} + ( 2 \beta_{1} + 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{33} + ( 6 - 8 \beta_{3} - 6 \beta_{4} - 8 \beta_{6} - 8 \beta_{7} ) q^{34} + ( -8 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} + 3 \beta_{6} - 5 \beta_{7} ) q^{35} + ( -3 + 3 \beta_{4} ) q^{36} + ( 6 \beta_{1} + 3 \beta_{6} - 3 \beta_{7} ) q^{37} + ( 8 \beta_{1} + 24 \beta_{2} + 4 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{38} + ( -2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{39} + ( -3 + 8 \beta_{1} + 12 \beta_{2} - 5 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} ) q^{40} + ( -26 - 12 \beta_{3} - 6 \beta_{6} - 6 \beta_{7} ) q^{41} + ( 8 \beta_{1} + 4 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{42} + ( -16 \beta_{1} - 12 \beta_{2} + 8 \beta_{6} - 8 \beta_{7} ) q^{43} + ( 46 + 2 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} ) q^{44} + ( 3 + 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{7} ) q^{45} + ( 16 - 4 \beta_{3} + 4 \beta_{4} ) q^{46} + ( 16 \beta_{1} - 4 \beta_{2} - 8 \beta_{6} + 8 \beta_{7} ) q^{47} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{5} - 7 \beta_{6} + 7 \beta_{7} ) q^{48} + ( -9 + 12 \beta_{3} + 6 \beta_{6} + 6 \beta_{7} ) q^{49} + ( 4 + 5 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} ) q^{50} + ( 6 \beta_{3} + 12 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{51} + ( 12 \beta_{1} + 4 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{52} + ( 6 \beta_{1} + 3 \beta_{6} - 3 \beta_{7} ) q^{53} + ( -3 + 3 \beta_{3} ) q^{54} + ( 18 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 15 \beta_{6} + 3 \beta_{7} ) q^{55} + ( -26 - 16 \beta_{3} - 6 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} ) q^{56} + ( -12 \beta_{1} - 6 \beta_{6} + 6 \beta_{7} ) q^{57} + ( -26 \beta_{1} + 12 \beta_{2} - 6 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{58} + ( -14 \beta_{3} + 4 \beta_{4} + 9 \beta_{6} + 9 \beta_{7} ) q^{59} + ( 3 - 4 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} ) q^{60} + 38 q^{61} + ( -16 \beta_{1} - 32 \beta_{2} + 16 \beta_{6} - 16 \beta_{7} ) q^{62} + ( 6 \beta_{1} + 12 \beta_{2} - 3 \beta_{6} + 3 \beta_{7} ) q^{63} + ( -5 + 16 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{64} + ( 20 + 2 \beta_{3} - 4 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{65} + ( -6 + 8 \beta_{3} + 6 \beta_{4} + 8 \beta_{6} + 8 \beta_{7} ) q^{66} + ( -28 \beta_{1} + 4 \beta_{2} + 14 \beta_{6} - 14 \beta_{7} ) q^{67} + ( 4 \beta_{1} - 36 \beta_{2} + 2 \beta_{5} - 14 \beta_{6} + 14 \beta_{7} ) q^{68} + ( -20 + 4 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{69} + ( -12 - 12 \beta_{1} - 28 \beta_{2} - 12 \beta_{3} - 8 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} ) q^{70} + ( -4 \beta_{3} - 24 \beta_{4} - 10 \beta_{6} - 10 \beta_{7} ) q^{71} + ( -6 \beta_{1} - 6 \beta_{2} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{72} + ( -4 \beta_{1} - 8 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -30 + 6 \beta_{4} ) q^{74} + ( -6 \beta_{1} + \beta_{2} - 8 \beta_{3} - 12 \beta_{4} + \beta_{6} - 5 \beta_{7} ) q^{75} + ( -12 + 32 \beta_{3} + 12 \beta_{4} + 8 \beta_{6} + 8 \beta_{7} ) q^{76} + ( -12 \beta_{1} + 8 \beta_{5} - 10 \beta_{6} + 10 \beta_{7} ) q^{77} + ( 4 \beta_{1} + 12 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{78} + ( -8 \beta_{3} - 8 \beta_{4} ) q^{79} + ( -42 + 6 \beta_{1} + 26 \beta_{2} + 16 \beta_{3} + 10 \beta_{4} - 5 \beta_{5} - \beta_{6} + 9 \beta_{7} ) q^{80} + 9 q^{81} + ( -26 \beta_{1} - 24 \beta_{2} + 12 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} ) q^{82} + ( -8 \beta_{1} + 20 \beta_{2} + 4 \beta_{6} - 4 \beta_{7} ) q^{83} + ( -22 + 6 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} ) q^{84} + ( 12 - 28 \beta_{1} - 18 \beta_{3} - 8 \beta_{5} - 19 \beta_{6} + \beta_{7} ) q^{85} + ( -36 - 12 \beta_{3} - 16 \beta_{4} ) q^{86} + ( 18 \beta_{1} - 14 \beta_{2} - 9 \beta_{6} + 9 \beta_{7} ) q^{87} + ( 36 \beta_{1} - 36 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{88} + ( 66 + 8 \beta_{3} + 4 \beta_{6} + 4 \beta_{7} ) q^{89} + ( -15 + 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{90} + ( -12 \beta_{3} - 8 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{91} + ( 16 \beta_{1} + 8 \beta_{5} ) q^{92} + ( 8 \beta_{1} - 8 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} ) q^{93} + ( 52 - 4 \beta_{3} + 16 \beta_{4} ) q^{94} + ( 12 \beta_{1} + 48 \beta_{2} + 16 \beta_{3} + 24 \beta_{4} - 2 \beta_{6} + 10 \beta_{7} ) q^{95} + ( 53 + 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{96} + ( 48 \beta_{1} + 8 \beta_{5} + 20 \beta_{6} - 20 \beta_{7} ) q^{97} + ( -9 \beta_{1} + 24 \beta_{2} - 12 \beta_{5} + 12 \beta_{6} - 12 \beta_{7} ) q^{98} + ( -6 \beta_{3} - 12 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 10q^{4} + 4q^{5} - 6q^{6} + 24q^{9} + O(q^{10})$$ $$8q - 10q^{4} + 4q^{5} - 6q^{6} + 24q^{9} - 42q^{10} + 20q^{14} - 46q^{16} + 52q^{20} + 72q^{21} - 18q^{24} + 32q^{25} + 84q^{26} - 184q^{29} - 60q^{30} + 12q^{34} - 30q^{36} - 6q^{40} - 256q^{41} + 348q^{44} + 12q^{45} + 112q^{46} - 24q^{49} + 72q^{50} - 18q^{54} - 244q^{56} + 6q^{60} + 304q^{61} - 10q^{64} + 168q^{65} - 12q^{66} - 144q^{69} - 104q^{70} - 252q^{74} - 24q^{76} - 308q^{80} + 72q^{81} - 204q^{84} + 24q^{85} - 280q^{86} + 560q^{89} - 126q^{90} + 376q^{94} + 426q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + \nu^{3} + 4 \nu$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + \nu^{4} + 4 \nu^{2} + 16$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} + 12 \nu^{3}$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 16 \nu^{4} + 24 \nu^{3} + 16 \nu^{2} + 80 \nu + 128$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 5 \nu^{5} + 16 \nu^{4} - 24 \nu^{3} + 16 \nu^{2} - 80 \nu + 128$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{2} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} + 2 \beta_{6} - \beta_{4} - 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} - \beta_{6} - \beta_{5} + 18 \beta_{2} - 2 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} + 16 \beta_{3} - 5$$ $$\nu^{7}$$ $$=$$ $$-13 \beta_{7} + 13 \beta_{6} - 19 \beta_{5} - 42 \beta_{2} - 22 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −1.52274 − 1.29664i −1.52274 + 1.29664i −0.656712 − 1.88911i −0.656712 + 1.88911i 0.656712 − 1.88911i 0.656712 + 1.88911i 1.52274 − 1.29664i 1.52274 + 1.29664i
−1.52274 1.29664i 1.73205 0.637459 + 3.94888i 4.27492 2.59328i −2.63746 2.24584i 0.837253 4.14959 6.83966i 3.00000 −9.87212 1.59414i
19.2 −1.52274 + 1.29664i 1.73205 0.637459 3.94888i 4.27492 + 2.59328i −2.63746 + 2.24584i 0.837253 4.14959 + 6.83966i 3.00000 −9.87212 + 1.59414i
19.3 −0.656712 1.88911i −1.73205 −3.13746 + 2.48120i −3.27492 3.77822i 1.13746 + 3.27203i −9.55505 6.74766 + 4.29756i 3.00000 −4.98678 + 8.66787i
19.4 −0.656712 + 1.88911i −1.73205 −3.13746 2.48120i −3.27492 + 3.77822i 1.13746 3.27203i −9.55505 6.74766 4.29756i 3.00000 −4.98678 8.66787i
19.5 0.656712 1.88911i 1.73205 −3.13746 2.48120i −3.27492 3.77822i 1.13746 3.27203i 9.55505 −6.74766 + 4.29756i 3.00000 −9.28814 + 3.70547i
19.6 0.656712 + 1.88911i 1.73205 −3.13746 + 2.48120i −3.27492 + 3.77822i 1.13746 + 3.27203i 9.55505 −6.74766 4.29756i 3.00000 −9.28814 3.70547i
19.7 1.52274 1.29664i −1.73205 0.637459 3.94888i 4.27492 2.59328i −2.63746 + 2.24584i −0.837253 −4.14959 6.83966i 3.00000 3.14704 9.49190i
19.8 1.52274 + 1.29664i −1.73205 0.637459 + 3.94888i 4.27492 + 2.59328i −2.63746 2.24584i −0.837253 −4.14959 + 6.83966i 3.00000 3.14704 + 9.49190i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.3.f.b 8
3.b odd 2 1 180.3.f.h 8
4.b odd 2 1 inner 60.3.f.b 8
5.b even 2 1 inner 60.3.f.b 8
5.c odd 4 2 300.3.c.f 8
8.b even 2 1 960.3.j.e 8
8.d odd 2 1 960.3.j.e 8
12.b even 2 1 180.3.f.h 8
15.d odd 2 1 180.3.f.h 8
15.e even 4 2 900.3.c.r 8
20.d odd 2 1 inner 60.3.f.b 8
20.e even 4 2 300.3.c.f 8
40.e odd 2 1 960.3.j.e 8
40.f even 2 1 960.3.j.e 8
60.h even 2 1 180.3.f.h 8
60.l odd 4 2 900.3.c.r 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.b 8 1.a even 1 1 trivial
60.3.f.b 8 4.b odd 2 1 inner
60.3.f.b 8 5.b even 2 1 inner
60.3.f.b 8 20.d odd 2 1 inner
180.3.f.h 8 3.b odd 2 1
180.3.f.h 8 12.b even 2 1
180.3.f.h 8 15.d odd 2 1
180.3.f.h 8 60.h even 2 1
300.3.c.f 8 5.c odd 4 2
300.3.c.f 8 20.e even 4 2
900.3.c.r 8 15.e even 4 2
900.3.c.r 8 60.l odd 4 2
960.3.j.e 8 8.b even 2 1
960.3.j.e 8 8.d odd 2 1
960.3.j.e 8 40.e odd 2 1
960.3.j.e 8 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 92 T_{7}^{2} + 64$$ acting on $$S_{3}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 80 T^{2} + 24 T^{4} + 5 T^{6} + T^{8}$$
$3$ $$( -3 + T^{2} )^{4}$$
$5$ $$( 625 - 50 T - 6 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$7$ $$( 64 - 92 T^{2} + T^{4} )^{2}$$
$11$ $$( 24576 + 348 T^{2} + T^{4} )^{2}$$
$13$ $$( 1536 + 84 T^{2} + T^{4} )^{2}$$
$17$ $$( 221184 + 1044 T^{2} + T^{4} )^{2}$$
$19$ $$( 221184 + 1008 T^{2} + T^{4} )^{2}$$
$23$ $$( 1024 - 368 T^{2} + T^{4} )^{2}$$
$29$ $$( 16 + 46 T + T^{2} )^{4}$$
$31$ $$( 393216 + 2304 T^{2} + T^{4} )^{2}$$
$37$ $$( 124416 + 756 T^{2} + T^{4} )^{2}$$
$41$ $$( -1028 + 64 T + T^{2} )^{4}$$
$43$ $$( 1364224 - 2528 T^{2} + T^{4} )^{2}$$
$47$ $$( 614656 - 3296 T^{2} + T^{4} )^{2}$$
$53$ $$( 124416 + 756 T^{2} + T^{4} )^{2}$$
$59$ $$( 69033984 + 16668 T^{2} + T^{4} )^{2}$$
$61$ $$( -38 + T )^{8}$$
$67$ $$( 7573504 - 9392 T^{2} + T^{4} )^{2}$$
$71$ $$( 884736 + 17136 T^{2} + T^{4} )^{2}$$
$73$ $$( 3538944 + 4176 T^{2} + T^{4} )^{2}$$
$79$ $$( 393216 + 2304 T^{2} + T^{4} )^{2}$$
$83$ $$( 2027776 - 4064 T^{2} + T^{4} )^{2}$$
$89$ $$( 3988 - 140 T + T^{2} )^{4}$$
$97$ $$( 495550464 + 45888 T^{2} + T^{4} )^{2}$$